Spheres and symmetry: Borsuk’s antipodal theorem. (English) Zbl 0795.55004

This nicely-written survey article concerns the fundamental discovery of Borsuk that a continuous function from the \(n\)-sphere to Euclidean space of the same dimension must send at least one pair of antipodal points of the sphere to the same point in the Euclidean space. The author not only lists eight equivalent, and impressively diverse, equivalent forms of Borsuk’s result, he also sketches three strikingly different approaches to proving it. A brief, interesting exploration of some of the limits to extending Borsuk’s theorem is followed by the discussion of a particularly fruitful direction of generalization: to group actions on spheres. After an exceedingly condensed section on index theories, the paper closes with eight well-chosen and clearly-described applications of Borsuk’s discovery. This relatively brief survey includes a useful 80- item list of references, but a reader who wishes a more detailed and comprehensive survey of the same topic should consult the author’s article in [Méthodes topologiques en analyse non linéaire, Sémin. Math. Supér., Sémin. Sci. OTAN (NATO Adv. Study Inst.) 95, 166-235 (1985; Zbl 0573.55003)].


55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)


Zbl 0573.55003
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